Connections on bundles
Higher abelian differential cohomology
Higher nonabelian differential cohomology
Application to gauge theory
The notion of connection on a 2-bundle generalizes the notion of connection on a bundle from principal bundles to principal 2-bundles / gerbes.
It comes with a notion of 2-dimensional parallel transport.
For an exposition of the concepts here see also at infinity-Chern-Weil theory introduction the section Connections on principal 2-bundles .
For a Lie 2-group, a connection on a -principal 2-bundle coming from a cocycle is a lift of the cocycle to the 2-groupoid of Lie 2-algebra valued forms
On trivial 2-bundles
When the underlying principal 2-bundle over a smooth manifold is topologically trivial, then the connections on it are identified with Lie 2-algebra valued differential forms on .
Recall from the discussion there what such form data looks like.
Let be some Lie 2-algebra. For instance for discussion of connections on -gerbes ( a Lie group) this would be the derivation Lie 2-algebra of the Lie algebra of .
Let and be the two vector spaces involved and let
be a dual basis, respectively. The structure of a Lie 2-algebra is conveniently determined by writing out the most general Chevalley-Eilenberg algebra
with these generators.
We thus have
for collections of structure constants (the bracket on ) and (the differential ) and (the action of on ) and (the “Jacobiator” for the bracket on ).
These constants are subject to constraints (the weak Jacobi identity and its higher coherence laws) which are precisely equivalent to the condition
Over a test space a -valued form datum is a morphism
from the Weil algebra .
This is given by a 1-form
and a 2-form
The curvature of this is , where the 2-form component (“fake curvature”) is
and whose 3-form component is
Differential Čech cocycle data
We spell out the data of a connection on a 2-bundle over a smooth manifold with respect to a given open cover , following (FSS, SchreiberCohesive)
Connections on 2-bundles with vanishing 2-form curvature and arbitrary 3-form curvature are defined in terms of their higher parallel transport are discussed in
Urs Schreiber, Konrad Waldorf,
Smooth Functors and Differential Forms, Homology, Homotopy Appl., 13(1), 143-203 (2011) (arXiv:0802.0663)
Connections on non-abelian gerbes and their holonomy, Theory and Applications of Categories, Vol. 28, 2013, No. 17, pp 476-540. (TAC, arXiv:0808.1923, web)
Examples of 2-connections with vanishing 2-form curvature obtained from geometric quantization are discusssed in
- Olivier Brahic, On the infinitesimal Gauge Symmetries of closed forms (arXiv)
The cocycle data for 2-connections with coeffcients in automorphism 2-groups but without restrictions on the 2-form curvature have been proposed in
- Paolo Aschieri, Luigi Cantini, Branislav Jurco, Nonabelian Bundle Gerbes, their Differential Geometry and Gauge Theory , Communications in Mathematical Physics Volume 254, Number 2 (2005) 367-400,(arXiv:hep-th/0312154).
A discussion of fully general local 2-connections is in
and the globalization is in
For a discussion of all this in a more comprehensive context see section xy of
See also connection on an infinity-bundle for the general theory.
Applications to physics
Nonabelian 2-connections appear for instance as orientifold B-fields in type II string theory, as differential string structure in heterotic string theory, and as fields in non-abelian 7-dimensional Chern-Simons theory. See at these pages for references.
An appearance in 4-dimensional Yang-Mills theory and 4d TQFT is reported in